Semigroups \\\\ A Note

Semigroups A Note Johar M. Ashfaque Abstract Semigroups can be used to solve a large class of problems commonly known as evolution equations. These types of equations appear in many fields including physics, chemistry and engineering. So, what are semigroups?

Definition 0.1 A semigroup is a pair (S, ·) where S is a non-empty set and · is the binary operation such that ·:S×S →S which is associative. Note. A semigroup unlike a group needs not to have an identity element nor an inverse element.

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Examples • Every group is a semigroup • The set of integers Z with multiplication • The set Mn (R) of n × n square matrices over a ring R with matrix multiplication – S = M2 (R)

A semigroup is commutative if the equation xy = yx holds for x, y ∈ S. Assume (S, ·) is a semigroup. Then • An element a ∈ S is idempotent if a · a = a • z ∈ S is a zero of S if x · z = z · x = z holds for all x ∈ S • An element n of S is an identity of S if n · x = x · n = x holds for all x ∈ S

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